B. Ellipses and Circles

Ellipse A = π ab a a = semi-major axis B = semi-minor axis b

Circle A circle is a set of points equidistant from a fixed point within the center. πd2 Cr d = 2r A= πr2 = = 4 2 C = 2πr = πd 2A = rReview of Plane FiguresC. Arcs and Sectors on CirclesImportant application of a radian measure-Stated in an elementary theorem in geometry“On a circle of radius r, a central angle ( an angle whose vertex is the centerof the circle) of θ radius intercepts an arc whose length is equal to theproduct of θ and r.”

s = rθ, θ in radiansEquivalent formulas θ = s/r , θ in radians r = s/θReview of Plane FiguresAreas of Sectors of CircleSector-a part of a circle between two radii with the given central angle

Area of a Circle A = πr2 = 1 ( 2π ) r2 , where 2π is the angle for the θ 2 complete circle r Area of a sector of a circle of radius r A = 1 r2θ 2 θ – denotes the central angle of the circle in radian measureReview of Plane FiguresD. Similar Figures If the corresponding angles in two figures are equal, it only means thatthey have the same shape though not necessarily of the same size. Suchrelation ships is called similar figures. Two segments are proportional if there exists a positive integerk, such that the length of one segment is k times the length of the other.

 The proportion property of two similar figures is that a

correspondence exists between the figures in such a way that everysegment in the first has a length that is k times that of its correspondingsegment in the other. Example: If length L1 = 12 and L2 = 18 then there L1 exists an integer k = 6 (common factor)such that the ratio of the two L2 lengths is 2:3.Review of Plane FiguresD. Similar FiguresExample:i. ΔABC ~ ΔA’B’Cii. rt. ΔCMN ~ rt. ΔCM’N’ C C

A’ B’ M’ N’

A B N M Two polygons are similar (~)  if their corresponding angles are equal and their corresponding sides are proportional.Review of Plane FiguresD. Similar Figures A Example: B’ E’ ABCDE ~ AB’C’D’E’B E i. ABC = AB’C’ C’ D’ BCD = B’C’D’ BAE = B’A’E’ C D ii. AB BC CD DE = = = AB’ B’C’ C’D’ D’E’Planes and Angles in Space A. Lines and Planes in Space B. Locus C. Dihedral and Polyhedral AnglesPlanes and Angles in SpaceA. Lines and Planes in SpacePlane- A surface such that a straight line joining any two points in it lieswholly in the surface- Understood to be indefinite in extent, but usually represented by aparallelogram lying in the plane- May be designated by a single small letter at one vertex or twocapital letters at opposite vertices

Postulate 2.1.1. Conditions for Existence of Plane

A plane is determined by any one of the following conditions: a. a line and a point not on the line b. three noncollinear points c. two intersecting lines d. two parallel linesPostulate 2.1.2. If two planes intersect, their intersection is a line.Planes and Angles in SpaceA. Lines and Planes in Space•Theorems on Perpendicular Lines and PlanesTheorem 2.1.1. If a line is perpendicular to each of two other lines at their point of intersection, it is perpendicular to the plane of the two lines.Theorem 2.1.2.All perpendiculars to a line at a point lie in the plane, which isperpendicular to the line at a given point.Theorem 2.1.3.At a given point on a line, only one plane perpendicular to the line can be drawn.Theorem 2.1.4.At a given external point, one and only one plane can be drawn perpendicular to a line.Theorem 2.1.5.Through a given point; there can be one and one plane perpendicular to a given plane.Corollary 2.1.1.The perpendicular line is the shortest line from a point to a plane.Planes and Angles in SpaceA. Lines and Planes in Space•Theorems on Parallel Lines and PlanesA line and a plane are parallel if they cannot meet, farthey are produced. Some essential theorems on parallel linesplanes are as follows:Theorem 2.1.6.Two lines perpendicular to the same plane are parallel.Theorem 2.1.7.Any plane containing one of two parallel line is parallel to theother.Theorem 2.1.8. If a line is parallel to a plane, the intersection of a plane with any plane passed through the given line is parallel to the line.Theorem 2.1.9.Two planes perpendicular to the same line are parallel.Theorem 2.1.10. If a third plane intersects each of two parallel planes, the lines intersection are parallel.Planes and Angles in SpaceA. Lines and Planes in Space•Theorems on Parallel Lines and Planes Theorem 2.1.11. The line perpendicular to one of two parallel planes is perpendicular to the other also.

Theorem 2.1.12. If two intersecting lines are parallel to a plane, the plane of these lines is parallel to that plane.

Theorem 2.1.13. If two angles not in the same plane have their sides respectively parallel and lying on the same side of the line joining their vertices, they are equal, and their planes are parallel.Planes and Angles in SpaceLocus Locus in three-dimensional space, as well as in two-dimensional plane is defined - geometric figure which contains only those points which satisfies certain conditions that contain all such points.

• Locus in Plane a. The locus of points at a given distance from given point is a circle having the given point as center and the distance as radius.

b. The locus of points at a given distance from

given line is a pair of lines parallel to the given line and at same distance from it.

c. The locus of points equidistant from two given

points is the perpendicular bisector of the line segment, joining the two points.Planes and Angles in SpaceLocus • Locus in Plane d. The locus of points equidistant from the sides of an angle is a line, which is the bisector of the angle

e. The locus of the vertex of a right triangle having a given hypotenuse is the circle having the hypotenuse as its diameter.Planes and Angles in SpaceLocus • Locus in Space a. The locus of points at a given distance from a given point is a sphere having the given point as a center and the distance as radius.

b. The locus of points at a given distance from a

given plane is a pair of planes parallel to a given plane at the same distance from it.Planes and Angles in SpaceLocus • Locus in Space c. The locus of points equidistant from two points is the plane, which is the perpendicular bisector of the line joining the two points.

d. The locus of points equidistant from two

intersecting planes is the plane, which is the bisector of the angle between them.Planes and Angles in SpaceLocus • Locus in Space e. The locus of the vertex of a right triangle having given hypotenuse is the sphere having the hypotenuse as its diameter.Planes and Angles in SpaceC. Dihedral and Polyhedral Angles • Dihedral Angle A - the opening between two intersecting planes. • Edge – the line of intersection CB of the plane D edge • Faces – the planes DC and AB face face

B C - Designated by its edge or by its two faces and its edge Thus, the dihedral angle of the figure is designated by A B A-BC-D, or when no confusion arises, simply by BC.

C DPlanes and Angles in SpaceC. Dihedral and Polyhedral Angles • Dihedral Angle A  The plane angle of a dihedral angle is the angle formed by two straight lines, one in each face, D perpendicular to the edge at the same point.

B C  Adjacent dihedral angles are dihedral angles which a have the same edge and a common face between them. Example: C-AB-D and D-AB-E are adjacent dihedral angles. C D B

A EPlanes and Angles in SpaceC. Dihedral and Polyhedral Angles • Dihedral Angle  Right dihedral angles -when one plane meets another to form equal adjacent dihedral angles  When one plane forms a right dihedral angle with one another, the planes are perpendicular to each other.

 Two Vertical dihedral angles

- are dihedral angles that have the same edge and the faces of one are prolongation of the faces of the other.Planes and Angles in SpaceC. Dihedral and Polyhedral Angles • Polyhedral Angle - A figure formed by three or more planes meeting at a common point • Faces – are the intersecting planes • Edges – are the lines of intersection of the faces • Vertex – is the point of intersection of the edges • Face angles – are the angles at the vertex formed by any two adjacent edges

• Dihedral angles of the polyhedral angle – are the dihedral angles formed by the intersecting faces

• Section – formed if a plane cuts all faces of the polyhedral angle (but not at the vertex) V F E